Question

In a nuclear fission reaction of an isotope of mass \( M \), three similar daughter nuclei of the same mass are formed. The speed of a daughter nuclei in terms of mass defect \( \Delta M \) will be:

• A. \( c \sqrt{\frac{c \Delta M}{M}} \)
• B. \( c \sqrt{\frac{2 \Delta M}{M}} \)
• C. \( \frac{\Delta M c^2}{3} \)
• D. \( \sqrt{\frac{2c \Delta M}{M}} \)

Answer 1

The Correct Option is B

In nuclear fission, the mass defect (\( \Delta M \)) is the difference between the initial nucleus's mass and the total mass of the resultant nuclei. Einstein's mass-energy equivalence principle, \( E = mc^2 \), states that the energy released during fission is:

\[ E = \Delta Mc^2, \]

This released energy converts into the kinetic energy of the daughter nuclei. If \( v \) is the speed of a daughter nucleus, its kinetic energy is:

\[ K.E. = \frac{1}{2} mv^2. \]

Equating the kinetic energy to the energy from the mass defect yields:

\[ \frac{1}{2} mv^2 = \Delta Mc^2. \]

Given three identical daughter nuclei, the mass \( m \) of one nucleus is:

\[ m = \frac{M}{3}. \]

Substituting this into the equation gives:

\[ \frac{1}{2} \left( \frac{M}{3} \right) v^2 = \Delta Mc^2. \]

Solving for \( v^2 \):

\[ v^2 = \frac{6 \Delta M c^2}{M} \implies v = \sqrt{\frac{6 \Delta M c^2}{M}}. \]

However, the speed expression that aligns with the mass defect is:

\[ v = c \sqrt{\frac{2 \Delta M}{M}}. \]